8: The Response to Phenotypic Selection

The Long-Term Response to Selection

If our selection pressure is sustained over many generations, we can use our breeder’s equation to predict the response. If we are willing to assume that our heritability does not change and we maintain a constant selection differential (\(S\)), then after \(n\) generations our phenotype mean will have shifted

\[n h^2 S\]

i.e. our population will keep up a linear response to selection.

Therefore, long-term, consistent selection can drive impressive evolutionary change. One example of this comes from a field experiment in Illinois, where plant breeders have systematically selected for higher and lower oil content in corn (see our previous Figure \(\PageIndex{5}\) for one generation of up selection). For over a century, they have taking seeds from the plants in the extremes of the distribution and using them to form the next generation. They have achieved impressive long-term responses, pushing the population distributions well beyond their initial range (Figure \(\PageIndex{8}\). For example, the oil up-selection line went from a mean oil content of \(4.7\%\) in 1896 to \(22.1\%\) in 2004. They’ve established two secondary populations where the selection differential was reversed. In the up-selection population they have maintained an impressively linear increase in oil content, shown by red line in Figure \(\PageIndex{7}\), but while the response is linear at first in the down line but they quickly reach very low oil content (limited by the physical boundary at 0% oil content).

In wild populations, selection pressures are likely rarely sustained for large numbers of generations. For example, the Grants’ have measured phenotypic selection in Darwin’s Finches over multiple decades on the island of Daphne Major. They have seen that selection pressures in the Medium ground-finch (Geospiza fortis) have reversed a number of times over the years (Figure \(\PageIndex{11}\)).

Patterns of long-term phenotypic change in the wild

Looking across the diversity of plants and animals we see huge changes in size and form, can the strengths of selection we can observe over short time periods possibly explain these changes?

To compare phenotypic changes over various time periods we need some measure of the rate of phenotypic change. proposed the rate of change from \(X_1\) to \(X_2\) in time interval \(\Delta t\), measured in millions of years, be quantified as

\[\frac{\log \left(\frac{X_2}{X_1} \right) }{\Delta t} = \frac{\log \left(X_2 \right) -\log \left(X_1 \right) }{\Delta t}\]

by expressing this the log of the ratio, we are looking at the proportional fold change, which makes sense as a evolutionary change of 1cm in length is more impressive if you’re a mouse than an elephant. By putting this on a \(\log\)-scale we are looking at the fold relative change. called the units of this measure `the Darwin’, with a one Darwin change corresponding to a \(e\approx 2.71\) fold change in a million years, a two Darwin change corresponding to a \(e^2\approx 7.34\) fold change in a million years and so on.

Calculate the rate of change in body size in the Jersey red deer from Question \ref{question:reddeer} in Darwins. Do the same for the total change in corn oil content in the up lines in Figure \ref{Fig:Illinois_LTS_means}.

examined the absolute rate of phenotypic change in field study data and the fossil record, a dataset considerably expanded by . In Figure \(\PageIndex{13}\) each point is an observation of phenotype evolution. The x-axis shows the time period in years over which the evolutionary change was observed, the x-axis is plotted on a \(\log_{10}\) scale. The y-axis shows absolute rate of phenotypic change, measured in Darwins, again on a \(\log_{10}\) scale.

Over short timescales we see incredibly rapid evolution, note the high rates on the left of Figure \(\PageIndex{13}\). For example, the first black dot from the left is a case of evolution over decades in dog whelks. The invasion the green crab (Carcinus maenas) drove the evolution of more robust shells in Atlantic dog whelk (Nucella lapillus) in response to predation along the North American coast . The shell lip thickness of dog whelks in the St. Andrews, New Brunswick population had changed from 0.94mm to 1.44mm in just 25 years. That’s a 50% increase, and a rate of 17060 Darwins.

However, when we observe phenotypic evolution over longer time periods it is usually much slower. For example, the rightmost black dot in Figure \(\PageIndex{13}\) shows the phenotypic evolution along the lineage leading to Triceratops. Triceratops measured in an impressive 25.9–29.5 ft in length. They evolved from a close relative of Protoceratops, which was a bit bigger than a sheep at \(\sim\)5.9 ft in about 7.5 million years . However, that’s only a phenotypic change of \(0.143\) Darwins, its only a roughly four fold change in millions of years. These rates of change in dinosaurs have nothing on our dog whelks, or many other examples of evolution on short time scales. Thus evolutionary changes we can observe over short timescales readily explain long term changes in quantitative phenotypes.

Fitness and the Breeder’s Equation.

So directional evolution occurs as selection drives a change in the mean phenotype within a generation. But precisely how does this relate to the natural-selection requirement that organisms vary in their fitness? Some different ways of formulating the Breeder’s equation give us insight into the conditions for directional selection and the relationship to fitness landscapes.

Fitness Gradients and linear regressions

To understand this in more detail let imagine that we calculate the linear regression of an individual \(i\)’s mean-centered phenotype (\(X_i\)) on fitness (\(W_i\)), i.e.

\[W_i \sim \beta X_i + \overline{w} \label{fitness_regression}\]

the best fitting slope of this regression (\(\beta\)), we’ll call it the ‘fitness gradient’, is given by

\[\beta = Cov(X, \frac{w(X)}{\overline{w}} )/ V_P \label{beta_covar}\]

i.e. the fitness gradient is the phenotype-fitness covariance divided by the phenotypic variance. Using this result we can rewrite the breeder’s equation as

\[R= V_A \beta \label{eqn:R_beta}\]

i.e. we’ll see a directional response to selection if there is a linear relationship of phenotype on fitness, and if there is additive genetic variance for the phenotype. As one example of a fitness gradient, in Figure \(\PageIndex{16}\) the lifetime reproductive success (LRS) of male Red Deer is plotted against the weight of their antlers. The red line gives the linear regression of fitness (LRS) on antler mass and the slope of this line is the fitness gradient (\(\beta\)).

Fisher’s fundamental theorem of natural selection

Finally how does the mean fitness of our population evolve? If we choose relative fitness to be our phenotype (\(X=\frac{w(X)}{\overline{w}}\)), then the response in fitness is

\[\begin{aligned} R &= \frac{V_A}{V_P} Cov \left(\frac{w(X)}{\overline{w}} , \frac{w(X)}{\overline{w}} \right) = \frac{V_A}{V_P} V_P \nonumber\\ &=V_A\end{aligned}\]

i.e. the response to selection is equal to the additive genetic variance for relative fitness. Or as Fisher put it

“The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.” - (pg 37)

Fisher called this ‘the fundamental theorem of natural selection’. Our proof here is just a sketch, and more formal approaches are needed to show it in generality. There has been much gnashing of teeth over exactly how broadly this result holds, and exactly what Fisher meant .

Directional Selection on Fitness Landscapes

One common metaphor when we talk about evolution is that of a population exploring an adaptive landscape with natural selection pushing a population towards higher fitness states corresponding to peaks in this landscape (see e.g. Figure \(\PageIndex{17}\)). found an evocative formulation of the Breeder’s equation which aids our intuition of phenotypic fitness landscapes. showed that, if the phenotype is normally distributed, the response to selection (\(R\)) could be written in terms of the gradient (derivative) of the mean fitness (\(\overline{w}\)) of the population as a function of the mean phenotype:

\[R = \frac{V_A}{\overline{w}} \frac{\partial \overline{w}}{\partial \bar{x}} \label{eqn:pheno_fitness_landscape} %V_A % \frac{\partial \log \left(\overline{w} \right)}{\partial \bar{z}}\]

What does this mean? Well \(\frac{V_A}{\overline{w}}\) is always positive, so the direction our population responds to selection is predicted by the sign of the derivative (see Appendix Section \ref{section:calculus} for more on derivatives). If increasing the mean phenotype of the population slightly would increase mean fitness (\(\frac{\partial \overline{w}}{\partial \bar{x}} >0\)) our population will respond that generation by evolving toward higher values of the trait (\(R>0\)), left panel of Figure \(\PageIndex{18}\). Conversely, if decreasing the population mean phenotype slightly would increase the mean fitness (\(\frac{\partial \overline{w}}{\partial \bar{x}} <0\)) the population will that generation evolve towards lower values of the phenotype (middle panel of Figure \(\PageIndex{18}\)). Thus, if selection pressures remain constant, we can think of the population as evolving on an adaptive landscape where the elevation is given by the population mean fitness. Natural selection operates on the basis of individual-level fitness, but as a result of this our population is increasing in its average fitness, i.e. our population is becoming better adapted. We’ll discuss the caveats of this hill-climbing interpretation below.

What happens when it reaches the top of a peak? Well at the top of a peak \(\frac{\partial \overline{w}}{\partial \bar{x}}=0\), as it is a local maximum, and so \(R=0\). Assuming that the relationship between fitness and phenotype stays constant, our population will stay at the top of the fitness peak. This view of natural selection does not imply that the population is evolving to the best possible state. Our population is just marching up the hill of mean fitness (end panel Figure \(\PageIndex{18}\)). However, this peak isn’t necessarily the highest fitness peak but simply whichever peak was closest. So our population can become trapped on a local, but not global peak of fitness (see, for example Figure \(\PageIndex{17}\)).

One dramatic example documenting adaptive evolution to a new fitness optimum is offered by a remarkable time-series of stickleback evolution from a fossil lake-bed in Nevada . In this lake the layers of sediment are laid down each year allowing a very detailed time series with over five thousand fossils measured. The time-series documents the evolution towards a new set of optimum phenotypes in the fifteen thousand years after the initial invasion of the lake by a heavily   stickleback species. In Figure \(\PageIndex{19}\) the population mean number of touching pterygiophores, the bones supporting the dorsal spines, through the fossil record (Figure \ref{fig:Stickleback_fossil}). Note how quickly the species evolves toward its new value, presumably a fitness optimum in their new environment, and the long subsequent time interval over which the population mean phenotype fluctuates about its new value.

fitted a model of a population adapting to a fitness landscape, with a single peak, to these time-series data. Their fitted fitness surface is shown in the lower panel of Figure \(\PageIndex{20}\) . The arrows show the moves that the population mean phenotype is making on this inferred fitness surface. The population initially takes large steps up toward the peak of this surface and subsequently fluctuates around the peak. Under the interpretation that there is a single stationary peak these fluctuations represent genetic drift randomly knocking the population off its optimum, with selection acting to restore the population towards this local optimum.

Issues with the interpretation of fitness landscapes.

In practice, fitness landscapes may not be constant. The environment may be constantly changing so our population is constantly forced to change to keep up with the fitness peak. Indeed our environment may change so quickly that our population cannot keep up with the peak. Our population is still trying to increase its mean fitness, to ‘adapt’, but the landscape itself is evolving. In the case of very rapid environmental change our population may slide further and further away the peak, and as a consequence its mean fitness decreases which may drive the population to extinction if our population drops below \(\overline{w}<1\) for long enough. The conditions for extinction are an active area of research in the field of ‘Evolutionary rescue’. More generally, for our fitness landscape result (eqn \ref{eqn:pheno_fitness_landscape}) to hold, and for us to be able to talk of our population attempting to evolve to higher mean fitness states, we need the fitness of our phenotypes to be independent of the frequency of other phenotypes in the population. (This independence allows us to assume that the fitness of individuals is not a function of the mean phenotype, as needed in eqn \ref{eqn:proof_landscape}). The assumption of frequency independence may not hold when there is competition between individuals, e.g. for resources or mates, as then the fitness of an individual depends on the strategies pursued by other individuals in the populations.

Stabilizing and Disruptive selection

Up to now we have just looked at directional selection, where selection acts to change the mean phenotype. However, we can also use quantitative genetic models to describe other modes of selection, extending from effects on the population mean the next natural step is to think about selection which acts on the population variance. Selection might act more strongly against individuals in the tails of the distribution, with those closer to the mean phenotype having higher fitness, which lowers the variance. Selection could also disfavour individuals close to the population mean, with individuals with extreme phenotypes having higher fitness, which acts to increase the variance of the population.

Directional selection occurs because of the covariance between our phenotype and fitness, eqn \ref{S_covar}. Just as expressing directional selection as a covariance allowed us to characterize directional selection as the linear relationship between fitness and phenotype, \(\beta\), we can summarize the variance reducing selection by including a quadratic term in the regression of fitness on phenotype

\[w_i \sim \beta x_i + \frac{1}{2} \gamma x_i^2 + \overline{w} \label{fitness_regression_stab}\]

This \(\gamma\), the coefficient of the quadratic term in our model, is the quadratic selection gradient: the covariance of fitness and the squared deviation from the phenotypic mean (\(\mu_{BS}\)), i.e.

\[\gamma = \frac{Cov\left(w(X), (X-\mu_{BS})^2 \right)}{V^2}\]

Our \(\gamma\) describes the curvature of the fitness surface around the mean. Values of \(\gamma<0\) are consistent with stabilizing selection, reducing the variance. While values of \(\gamma>0\) are consistent with disruptive selection, increasing the variance.

Under stabilizing selection the individuals with extreme phenotypes in either tail have lower fitness, the result of which is to reduce the phenotypic variance within a generation. A classic case of stabilizing selection is birth weight in humans . Mary Karn collected data for nearly fourteen thousand pregnancies from 1935-46 for birth weight and mortality. These data are replotted in Figure \(\PageIndex{22}\). The variance of all births is \(1.575\)lb\(^2\), while in live births this was reduced to \(1.26\)lb\(^2\), a 20% reduction in variance due to stabilizing selection. It is worth noting that this selection pressure has been greatly reduced over the decades in societies with access to good prenatal care .

In Central Africa, Black-bellied seedcrackers ( Pyrenestes ostrinus) show disruptive selection on a remarkable beak-size polymorphism (Figure \ref{Black_bellied_seedcrackers_beaks}). The small-beaked individuals feed on soft seeds from one species of marsh sedge while the big-beaked individuals feed on hard seeds from another sedge, which requires ten times the force to crack. recorded the fates of hundreds of juveniles, and found that individuals with intermediate beak sizes survived at much lower rates (Figure \(\PageIndex{23}\)) because they were not well adapted to either seed resource. Break length is subject to disruptive selection, as can also be seen by the significant negative quadratic term in the regression of survival probability on break length. The variance of mandible length in the total sample of individuals was \(0.5\)mm\(^2\) in the survivors this variance increased by a factor of \(2.5\) to \(1.3\)mm\(^2\).

To illustrate how directional selection and quadratic terms play off during adaptation, let’s consider the goldenrod gall fly (Eurosta solidaginis), aka the goldenrod ball gallmaker. See Figure \(\PageIndex{25}\). As it’s wonderful name implies this insect lays its eggs in goldenrod plants, and the larvae release chemicals forcing the plant to form a gall that forms a home for the larvae as they develop. While this seems like a pretty sweet deal for the larvae, it is not without its perils.

When the small, ball galls fall prey to parasitism from parasitoid wasps. When all the ball galls are small in the population selection drives strong positive directional selection on gall size, with little stabilizing selection. Notice in the left panel of Figure \ref{gall_size_stab} the good agreement between the linear selection gradient and the fit including a linear and quadratic term. However, bigger galls fall under the pall of predation from downy woodpeckers and black-capped chickadees, who seek out the tasty larvae. Thus intermediate size galls are favoured, a fitness peak that the population quickly reaches. Once on this peak, as shown in the right panel of Figure \(\PageIndex{25}\) there is no directional selection, i.e. no linear slope, but there is strong stabilizing selection, i.e. a quadratic term. Thus the population will be maintained at this fitness peak indefinitely if the environment remains unchanged.

Summary 

• Phenotypic natural selection requires variation in phenotypes impacts fitness in a non-random way. For evolution by natural selection to occur we need this phenotypic variation to be heritable.
• A simple model for the response to directional selection on a phenotype is given by the breeder’s equation. The expected response to selection between adjacent generations (\(R\)) is proportional to the response to selection within a generation, the selection differential (\(S\)), where the constant of proportionality is the narrow-sense heritability (\(h^2\)) of the trait. Thus we expect to see, and indeed do see, strong responses to selection when selection causes large changes within a generation in heritable phenotypes.
• If selection pressures and heritability remain constant we expect a linear response to selection across many generations. We can often see this in selection experiments, but in the wild selection pressures often fluctuate from generation to generation. The large changes in phenotype we see in the fossil record are easily explained by the strengths of selection we see acting over short time-scales.
• There are two other common ways to write the breeder’s equation. The first uses the selection gradient (\(\beta\)), the regression of fitness on phenotype. The second, the fitness landscape interpretation relies on writing this selection gradient as the derivative of mean fitness by phenotype. The fitness landscape form to the breeder’s equation helps us to understand how and when we can expect selection to act to increase the mean fitness of the population.
• We can understand other forms of selection on our phenotype that are expected to act on the phenotypic variance rather than the mean, such as disruptive and stabilizing selection, by extensions of the breeder’s equation to include terms for quadratic selection gradients.
• You are studying the rapid evolution of light organ size in fireflies (Photinus pyralis) in response to light pollution on a prairie in Ohio. In January of 1985, a highway was constructed through the prairie with bright streetlights. Since fireflies use light signals to locate mates, individuals with smaller, and thus less visible, light organs were less successful at mating in these new light conditions. You know the light organ was, on average, 4mm long prior to the construction of the highway. In 2005, the average light organ size in this population before mating was 6mm. If this firefly has 1 generation per year and the narrow sense heritability is 0.1, what was the mean light organ length of successfully reproducing individuals in 1985 (the first year of selection)?
• You are a rabbit breeder, and you decide that you want rabbits with long fur. The phenotypic variance is \(4\textrm{cm}^2\). The covariance of fur length between between full sibs is \(1\textrm{cm}^2\). The mean fur length in the initial population is \(3\textrm{cm}\). You choose to breed the top \(\frac{1}{3}\) of the population with the longest fur, and their mean fur length is \(5\)cm.
• Assuming that the covariance between sibs is due to only additive genetic variance, how many generations of this selection regime will it take for the fur length to be \(10cm\) in the population?